Bivariate polynomial multiplication

We study the multiplicative complexity and the rank of the multiplication in the local algebras R_m,n=k[x,y]/(x^m+1 ,yⁿ⁺¹) and T_n=k[x,y]/(xⁿ⁺¹,xⁿy,...,yⁿ⁺¹ ) of bivariate polynomials. We obtain the lower bounds (21/3-0(1))·dim R_m,n, and (2½-0(1))·dim T_n for the multiplicative complexity of the multiplication in R_m,n and T_n, respectively. On the other hand, we derive the upper bounds 3·dim T_n-2n-2 and 3·dim R_m.n-m-n-3 for the rank of the multiplication in T_n and R_m,n, respectively, provided that the ground field k admits “fast” univariate polynomial multiplication mod x^N-1. Our results are also applicable to arbitrary finite dimensional algebras of truncated bivariate polynomials k[x,y]/I, where the ideal I=(x(d₀+1),x(d₁+1)y,...,x(d_n+1)yⁿ ,yⁿ⁺¹) is described by a degree pattern d₀⩾d₁⩾···⩾d_n ⩾0